Question

goal: prove the triangle sum theorem. triangle sum theorem: the three angle measures of a triangle add to 180°. goal: to prove m∠1 + m∠2 + m∠3 = 180° diagram of triangle abc with line xy through a, angles 4,1,5 at a and angles 2 at b, 3 at c table with statement and justification columns

Explanation:

Step1: Draw line XY through A

Construct line $XY \parallel BC$ passing through vertex $A$.

Step2: Identify angle relationships

$\angle 4 = \angle 2$ (alternate interior angles, $XY \parallel BC$), $\angle 5 = \angle 3$ (alternate interior angles, $XY \parallel BC$).

Step3: Use straight angle property

$\angle 4 + \angle 1 + \angle 5 = 180^\circ$ (straight angle on line $XY$).

Step4: Substitute equal angles

Replace $\angle4$ with $\angle2$ and $\angle5$ with $\angle3$: $\angle2 + \angle1 + \angle3 = 180^\circ$.

Answer:

The triangle sum theorem is proven: $m\angle1 + m\angle2 + m\angle3 = 180^\circ$

The completed table is:

STATEMENT	JUSTIFICATION
$\angle4 = \angle2$, $\angle5 = \angle3$	Alternate Interior Angles Theorem
$\angle4 + \angle1 + \angle5 = 180^\circ$	Straight Angle Definition
$\angle2 + \angle1 + \angle3 = 180^\circ$	Substitution Property